Measurement
Class 7 | Science — Physics | Unit 1
Class 7 Unit 1 Science
UNIT 1 MEASUREMENT
Picture shows the relationship between volume, density and mass. 3D solid shapes sphere, cylinder, cone, cube, cuboid, hexagonal is given.
Learning Objectives
After studying this unit, students will be able to:
- identify fundamental and derived physical quantities.
- identify fundamental and derived units.
- obtain units for certain derived quantities.
- measure the area and volume of some regular shaped and irregular shaped objects.
- convert the volume of objects from cubic metre to litre and vice versa.
- calculate the density of solids and liquids.
- define astronomical unit and light year.
Introduction
In day to day life, we measure many things such as weight of fruits, vegetables and food grains, volume of liquids, temperature of the body, speed of the vehicles etcetera. Quantities such as mass, weight, distance, temperature, volume are called physical quantities. A value and a unit are used to express the magnitude of a physical quantity. For example, let us assume that you walk 2 kilometre every day. In this example '2' is the value and 'kilometre' is the unit used to express the magnitude of distance which is a physical quantity. In this lesson, we are going to study about fundamental quantities, derived quantities such as area, volume and density, and measurement of larger quantities.
1.1 Fundamental Quantities and Derived Quantities
Generally, physical quantities are classified into two types. They are: fundamental quantities and derived quantities.
1.1.1 Fundamental Quantities
A set of physical quantities which cannot be expressed in terms of any other quantities are known as fundamental quantities. Example: Length, Mass, Time. Their corresponding units are called fundamental units. There are seven fundamental physical quantities in SI Units (System of International Units). They are given in Table 1.1.
Table 1.1 Fundamental quantities and their units | Fundamental quantity | Fundamental unit |
|---|
| Length | metre (m) |
| Mass | kilogram (kg) |
| Time | second (s) |
| Temperature | Kelvin (K) |
| Electric current | Ampere (A) |
| Amount of substance | mole (Mol) |
| Luminous intensity | Candela (cd) |
1.1.2 Derived quantities
All other physical quantities which can be obtained by multiplying, dividing or by mathematically combining the fundamental quantities are known as derived quantities. Example: Area and volume. Their corresponding units are called derived units. Some of the derived quantities and their units are given in Table 1.2.
Table 1.2 Derived quantities and their units | Derived quantity | Unit |
|---|
| Area = Length × Breadth | Square metre (m²) |
| Volume = Length × Breadth × Height | m³ |
| Speed = Distance ÷ Time | ms⁻¹ |
| Electric Charge = Electric Current × Time | C (coulomb) |
| Density = Mass ÷ Volume | Kilogram m⁻³ |
1.2 Area
Area is a measure of how much space is there on a flat surface. The area of a plot of land is derived by multiplying its length and breadth.
Area = length × breadth
The unit of the area is m² (Read as square metre). Area is a derived quantity as we obtain it by multiplying the fundamental physical quantity length (length × breadth).
Problem 1.1
What is the area of 10 squares each having side of 1 metre?
Solution:
Area of a square = side × side = 1 metre × 1 metre = 1 m² or 1 square metre
Area of 10 squares = 1 square metre × 10 = 10 square metre
1.2.1 Area of regularly shaped objects
The area of regularly shaped objects can be calculated using the relevant formulae. In Table 1.3, the formulae used to calculate the area of certain regularly shaped figures are given.
Table 1.3 Area of some regularly shaped objects | Serial Number | Plane figure | Diagram | Area |
|---|
| 1 | Square | Picture of a square with sides marked as a | side × side = a × a = a² |
| 2 | Rectangle | Picture of a rectangle with sides marked as l and b | length × breadth = l × b = lb |
| 3 | Circle | Picture of a circle with radius marked as r | π × (radius)² = π × r² = πr² |
| 4 | Triangle | Picture of a triangle with sides marked h and b | ½ × base × height = ½ × b × h |
Problem 1.2
Find the area of the following regular shaped figures (Take π = 22/7).
A. A rectangle whose length is 12 metre and breadth is 4 metre.
B. A circle whose radius is 7 metres.
C. A triangle whose base is 6 metre and height is 8 metre.
Solution:
A. Area of rectangle = length × breadth = 12 × 4 = 48 square metre
B. Area of circle = π × r² = (22/7) × 7 × 7 = 154 square metre
C. Area of triangle = ½ × base × height = ½ × 6 × 8 = 24 square metre
1.2.2 Area of irregularly shaped objects
In our daily life, we encounter many irregularly shaped objects like leaves, maps, stickers of stars or flowers, peacock feather etc. The area of such irregularly shaped objects cannot be calculated using any formula. How can we find the area of these irregularly shaped objects? We can find the area of these figures with the help of a graph sheet. The following activity shows how to find the area of irregularly shaped plane figures.
Activity 1
Take a leaf from any one of the trees. Place it on a graph sheet and draw the outline of the leaf with a pencil. Remove the leaf. You can see the outline of the leaf on the graph sheet.
1. Now, count the number of whole squares enclosed within the outline of the leaf. Take it as M.
2. Then, count the number of squares that are more than half. Take it as N.
3. Next, count the number of squares which are half of a whole square. Note it as P.
4. Finally, count the number of squares that are less than half. Let it be Q.
Now, the approximate area of the leaf can be calculated using the following formula.
Approximate area of the leaf = M + (3/4)N + (1/2)P + (1/4)Q square centimetre.
Area of the leaf = centimetre².
Activity 2
Draw the following regularly shaped figures on a graph sheet and find their area by the graphical method. Also, find their area using appropriate formula. Compare the results obtained in two methods by tabulating them.
a. A rectangle whose length is 12 centimetre and breadth is 4 centimetre.
b. A square whose side is 6 centimetres
c. A circle whose radius is 7 centimetres
d. A triangle whose base is 6 centimetre and height is 8 centimetre.
| Serial Number | Shape | Area using formula | Area using graphical method |
|---|
| 1 | Rectangle | | |
| 2 | Square | | |
| 3 | Circle | | |
| 4 | Triangle | | |
Do You Know?
One square metre is the area enclosed inside a square of side 1 metre. Even though area is given in square metre, the surface need not to be square in shape.
1.3 Volume
The amount of space occupied by a three-dimensional object is known as its volume.
Volume = Surface area × Height
The International System of Unit (SI) unit of volume is cubic metre or m³.
1.3.1 Volume of regularly shaped objects
As in the case of area, the volume of a regularly shaped objects can also be determined using an appropriate formula. Table 1.4 gives the formulae used to calculate the volume of the regularly shaped objects.
Table 1.4 Volume of regularly shaped objects | Serial Number | Object | Figure | Volume |
|---|
| 1 | Cube | Picture of a cube with sides marked as a | side × side × side = a × a × a = a³ |
| 2 | Cuboid | Picture of a cuboid with sides marked as l, b, h | length × breadth × height = l × b × h = lbh |
| 3 | Sphere | Picture of a sphere with radius marked as r | 4/3 × π × (radius)³ = 4/3 × π × r³ = 4/3 πr³ |
| 4 | Cylinder | Picture of a cylinder with radius and height marked as r and h | π × (radius)² × height = π × r² × h = πr²h |
Problem 1.3
Find the volume of the following (Take π = 22/7).
a. A cube whose side is 3 centimetres.
b. A cylinder whose radius is 3 metre and height is 7 metre
Solution:
a. Volume of a cube = side × side × side = 3 cm × 3 cm × 3 cm = 27 cubic centimetre or cm³
b. Volume of a cylinder = π × (radius)² × height = 22/7 × 3 × 3 × 7 = 198 cm³
1.3.2 Volume of Liquids
Liquids also occupy some space and hence they also have volume. But, liquids do not possess any definite shape. So, the volume of a liquid cannot be determined as in the case of solids. When a liquid is poured into a container, it takes the shape and volume of the container. The volume of any liquid is equal to the space that it fills and it can be measured using a measuring cylinder or measuring beaker. The maximum volume of liquid that a container can hold is known as the capacity of the container.
A measuring container is graduated as shown in figure. Four measuring containers are given.
The volume of a liquid is equal to the volume of space it fills in the container. This can be directly observed from the readings marked in the measuring containers. If we notice the measuring cups given in figure carefully, we can observe that the readings are marked in the unit of 'ml'. This actually represents millilitre. To understand this unit of volume, let us first understand how much a litre means.
Litre is the commonly used unit to measure the volume of liquids. We know that the unit of volume is cubic centimetre if the dimensions of the object are given in centimetre. This cubic centimetre is commonly known as 'cc'. A volume of 1000 cm³ is termed as one litre (l).
1 litre = 1000 cm³ or cm³
1000 milli litre = 1 litre
1.3.3 Volume of irregularly shaped objects
There is no formula to determine the volume of irregularly shaped objects as in the case of area. For such objects, volume can be determined using a measuring cylinder and water.
Activity 3
Take a measuring cylinder and pour some water into it (Do not fill the cylinder completely). Note down the volume of water from the readings of the measuring cylinder. Take it as V₁.
Now take a small stone and tie it with a thread. Immerse the stone inside the water by holding the thread. This has to be done such that the stone does not touch the walls of the measuring cylinder. Now, the level of water will raise. Note down the volume of water and take it as V₂.
The volume of the stone is equal to the raise in the volume of water.
Volume of stone = V₂ - V₁
Do You Know?
To measure the volume of liquids, some other units are also used. Some of them are gallon, ounce, and quart.
1 gallon = 3785 milli litre
1 ounce = 30 milli litre
1 quart = 1 litre
1.4 Density
Take water in a beaker and drop an iron ball and a cork into the water. What do you observe? The iron ball sinks and the cork floats as shown in figure. Can you explain why?
Picture shows that iron ball sinks while cork floats in water.
If your answer is heavy objects sink in water and lighter objects float in water, then, why does a metal coin sinks in water whereas a much heavier wooden log floats? These questions can be answered if we understand the concept of density.
Picture shows that the lighter coin sinks while heavier wooden log floats.
Activity 4
a. Take an iron block and a wooden block of same mass (say 1kilogram each). Measure their volume. Which one has more volume and occupies more volume?
Answer:
b. Take an iron block and a wooden block of same size. Weigh them and measure their mass. Which one of them has more mass?
Answer:
From activity 4, we observe that wooden block occupies more volume than the iron ball of same mass. Also, we observe that wooden block is lighter than the iron block of same size. The lightness or heaviness of a body is due to density. If more mass is packed into some volume, it has greater density. So, the iron block will have more mass than the wooden block of the same size. Therefore, iron has more density.
Density of a substance is defined as the mass of the substance contained in unit volume (1 m³). If the mass of a substance is M and volume is V, then, its density is given as:
Density D = Mass (M) ÷ Volume (V)
D = M ÷ V
International Standard unit of density is kilogram ÷ m³. The Centimeter Gram Second System unit of density is gram ÷ cm³.
1.4.1 Density of different materials
Different materials have different densities. The materials with more density are called denser and the materials with less density are called rarer. The density of some widely used materials are listed in Table 1.5.
Table 1.5 Density of some common substances, at room temperature | Serial Number | Nature | Materials | Density (kg/m³) |
|---|
| 1 | Gas | Air | 1.2 kg/m³ |
| 2 | Liquid | Kerosene | 800 kg/m³ |
| 3 | Liquid | Water | 1000 kg/m³ |
| 4 | Liquid | Mercury | 13600 kg/m³ |
| 5 | Solid | Wood | 770 kg/m³ |
| 6 | Solid | Aluminium | 2700 kg/m³ |
| 7 | Solid | Iron | 7800 kg/m³ |
| 8 | Solid | Copper | 8900 kg/m³ |
| 9 | Solid | Silver | 10500 kg/m³ |
| 10 | Solid | Gold | 19300 kg/m³ |
The relationship between mass, density and volume are represented in the following density triangle. Picture shows the relationship between mass, volume, density are represented in a triangle.
Density = Mass ÷ Volume
Mass = Density × Volume
Volume = Mass ÷ Density
Problem 1.4
A solid cylinder of mass 280 kilogram has a volume of 4 m³. Find the density of cylinder.
Solution:
Density of cylinder = Mass of cylinder ÷ Volume of cylinder = 280 ÷ 4 = 70 kg/m³
Problem 1.5
A box is made up of iron and it has a volume of 125 cm³. Find its mass if the density of iron is 7.8 g/cm³.
Solution:
Density = Mass ÷ Volume
Hence, Mass = Volume × Density = 125 × 7.8 = 975 grams.
Do You Know?
Water has more density than oils like cooking oil and castor oil, although these oils appear to be denser than water. Density of castor oil is 961 kg/m³. If we put one drop of water in oil, water drop sinks. But, if we put one drop of oil in water, oil floats and forms a layer on water surface. However, some oils are denser than water.
Problem 1.6
A sphere is made from copper whose mass is 3000 kilogram. If the density of copper is 8900 kilogram per m³, find the volume of the sphere.
Solution:
Density = Mass ÷ Volume
Hence, Volume = Mass ÷ Density = 3000 ÷ 8900 = 30/89 = 0.34 m³
1.5 Measuring larger distances
Normally, we use centimetre, metre and kilometre to express the distances that we measure in our day-to-day life. But, for space research, astronomers need to measure very long distances such as the distance between the earth and a star or the distance between two stars. To express these distances, we shall learn about two such units, namely:
1. Astronomical unit
2. Light year
1.5.1 Astronomical Unit
We all know that the earth revolves around the sun in an elliptical orbit. Hence, the distance between the sun and the earth varies every day. When the earth is in its perihelion position (the position when the distance between the Earth and the Sun is short), the distance between the earth and the sun is about 147.1 million kilometre. When the earth is in its aphelion position, (the position when the distance between Earth and the Sun is the largest) the distance is 152.1 million kilometre. The average distance between the earth and the sun is about 149.6 million kilometre. This average distance is taken as one astronomical unit. Neptune is 30 Astronomical Unit away from the Sun. It means it is thirty times farther than the Earth.
One astronomical unit is defined as the average distance between the earth and the sun.
1 Astronomical Unit = 149.6 million kilometre = 149.6 × 10⁶ kilometre = 1.496 × 10¹¹ metre
Picture shows the astronomical unit perihelion and aphelion position of earth is given in elliptical orbit.
1.5.2 Light year
The nearest star to our solar system is Proxima Centauri. It is at a distance of 268,770 Astronomical Unit. We can note here that using Astronomical Unit for measuring distances of stars would be unwieldy. Therefore, astronomers use a special unit, called 'light year', for measuring the distance in deep space.
We have learnt that the speed of light in vacuum is 3 × 10⁸ metre per second. This means that light travels a distance of 3 × 10⁸ metre in one second. In a year (non-leap), there are 365 days. Each day has 24 hours, each hour has 60 minutes and each minute has 60 seconds. Thus, the total number of seconds in one year = 365 × 24 × 60 × 60 = 3.153 × 10⁷ second
If light travels at a distance of 3 × 10⁸ metre in one second, then the distance travelled by light in one year = 3 × 10⁸ × 3.153 × 10⁷ = 9.46 × 10¹⁵ metre.
This distance is known as one light year.
One light year is defined as the distance travelled by light in vacuum during the period of one year.
1 Light year = 9.46 × 10¹⁵ metre.
In terms of light year, Proxima Centauri is at 4.22 light-years from Earth and the Solar System. The Earth is located about 25,000 light years away from the galactic centre.
Points to Remember
- A set of physical quantities which cannot be expressed in terms of any other quantities are known as fundamental quantities. Their corresponding units are called fundamental units.
- The physical quantities which can be obtained by mathematically combining (that is multiplying and dividing) the fundamental quantities are known as derived quantities. Their corresponding units are called derived units.
- The area of a figure is the region covered by the boundary of the figure. Its SI unit is square metre or m².
- The area of irregularly shaped figures can be calculated with the help of a graph sheet.
- The amount of space occupied by a three dimensional object is known as its volume. The SI unit of volume is cubic metre or m³.
- The volume of liquids are expressed in terms of litre. One litre = 1000 cubic centimetre
- The maximum volume of a liquid that a container can hold is known as the capacity of the container.
- Density of a substance is defined as the mass of the substance contained in unit volume (1 m³). SI unit of density is kilogram per m³. The CGS unit of density is gram per cm³. 1 gram per cm³ = 10³ kilogram per m³
- The materials with higher density are called denser materials and the materials with lower density are called rarer materials.
- If the density of a solid is higher than that of a liquid, it sinks in that liquid. If the density of a solid is lower than that of a liquid, it floats in that liquid.
- Density = Mass ÷ Volume; Mass = Density × Volume; Volume = Mass ÷ Density
- One astronomical unit is defined as the average distance between the Earth and the Sun. 1 astronomical unit = 149.6 × 10⁶ kilometre = 1.496 × 10¹¹ metre.
- One light year is defined as the distance travelled by light in vacuum during the period of one year. 1 Light year = 9.46 × 10¹⁵ metre
Evaluation
Exercise 1. Choose the best answer
1. Which of the following is a derived quantity?
a) mass b) time c) area d) length
2. Which of the following is correct?
a) 1 Litre = 1 cubic centimetre b) 1 Litre = 10 cubic centimetre c) 1 Litre = 100 cubic centimetre d) 1 Litre = 1000 cubic centimetre
3. SI unit of density is
a) kilogram per metre square b) kilogram per metre cube c) kilogram per metre d) gram per metre cube
4. Two spheres have mass and volume in the ratio 2:1. The ratio of their density is
a) 1:2 b) 2:1 c) 4:1 d) 1:4
5. Light year is the unit of
a) distance b) time c) density d) Both length and time Exercise 2. Fill in the blanks
1. Volume of irregularly shaped objects are measured using the law of .
2. One cubic metre is equal to cubic centimetre.
3. Density of mercury is .
4. One astronomical unit is equal to .
5. The area of a leaf can be measured using a .
Exercise 3. True or False 1. The region covered by the boundary of a plane figure is called its volume 2. Volume of liquids can be found using measuring containers. 3. Water is denser than kerosene. 4. A ball of iron floats in mercury. 5. A substance which contains less number of molecules per unit volume is said to be denser. Exercise 4. Match the following
a.
| 1. Area | |
| 2. Distance | |
| 3. Density | |
| 4. Volume | |
| 5. Mass | |
Options: a) light year, b) metre cube, c) metre square, d) kilogram, e) kilogram per metre
b.
| 1. Area | |
| 2. Length | |
| 3. Density | |
| 4. Volume | |
| 5. Mass | |
Options: a) gram per centimetre cube, b) measuring jar, c) amount of a substance, d) rope, e) plane figures
Exercise 5. Arrange the following in correct sequence 1 Litre 100 cubic centimetre 10 Litre 10 cubic centimetre Copper Aluminium Gold Iron Exercise 6. Use the analogy to fill in the blank
1. Area is to metre square as Volume is to .
2. Liquid is to Litre as Solid is to .
3. Water is to Kerosene as is to Aluminium.
Exercise 7. Consider the following statements and choose the correct option
1. Assertion: Volume of a stone is found using a measuring cylinder.
Reason: Stone is an irregularly shaped object.
a) Both assertion and reason are true and reason is the correct explanation of assertion. b) Both assertion and reason are true, but reason is not the correct explanation of assertion. c) Assertion is true but reason is false. d) Assertion is false but reason is true.
2. Assertion: Wood floats in water.
Reason: Water is a transparent liquid.
a) Both assertion and reason are true and reason is the correct explanation of assertion. b) Both assertion and reason are true, but reason is not the correct explanation of assertion. c) Assertion is true but reason is false. d) Assertion is false but reason is true.
3. Assertion: Iron ball sinks in water.
Reason: Water is denser than iron.
a) Both assertion and reason are true and reason is the correct explanation of assertion. b) Both assertion and reason are true, but reason is not the correct explanation of assertion. c) Assertion is true but reason is false. d) Assertion is false but reason is true. Exercise 8. Answer very briefly
1. Name some of the derived quantities.
2. Give the value of one light year.
3. Write down the formula used to find the volume of a cylinder.
4. Give the formula to find the density of objects.
5. Name the liquid in which iron ball sinks.
6. Name the units used to measure the distance between celestial objects.
7. What is the density of gold?
Exercise 9. Answer briefly
1. What are derived quantities?
2. Distinguish between the volume of liquid and capacity of a container.
3. Define the density of objects.
4. What is one light year?
5. Define - Astronomical unit
Exercise 10. Answer in detail
1. Describe the graphical method to find the area of an irregularly shaped plane figure.
2. How will you determine the density of a stone using a measuring jar?
Exercise 11. Questions based on Higher Order Thinking Skills
There are three spheres A, B, C as shown below. Sphere A and B are made of same material. Sphere C is made of a different material. Spheres A and C have equal radii. The radius of sphere B is half that of A. Density of A is double that of C.
Picture shows pictures of the spheres a, b, c.
Now answer the following questions.
1. Find the ratio of masses of spheres A and B.
2. Find the ratio of volumes of spheres A and B.
3. Find the ratio of masses of spheres A and C.
Exercise 12. Numerical problems
1. A circular disc has a radius 10 cm. Find the area of the disc in metre square (Use π = 3.14).
2. The dimension of a school playground is 800 metre × 500 metre. Find the area of the ground.
3. Two spheres of same size are made from copper and iron respectively. Find the ratio between their masses (Density of copper is 8,900 kg/m³ and iron is 7,800 kg/m³).
4. A liquid having a mass of 250 gram fills a space of 1000 cm³. Find the density of the liquid.
5. A sphere of radius 1 centimetre is made from silver. If the mass of the sphere is 33 gram, find the density of silver (Take π = 3.14)
Exercise 13. Cross word puzzle
Cross Word Puzzle Box is given.
Clues – Across
1. SI unit of temperature; 2. A derived quantity; 3. Mass per unit volume; 4. Maximum volume of liquid a container can hold
Clues – Down
a. A derived quantity; b. SI unit of volume; c. A liquid denser than iron; d. A unit of length used to measure very long distances
Answer
1. Kelvin; 2. Volume; 3. Density; 4. Capacity; a. Velocity; b. Cubic metre; c. Mercury; d. Lightyear
ICT CORNER
Measurement
Let's know about the effects of mass and volume on density
PROCEDURE:
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Step 2: Select the options at top right side window to customize
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Measurement URL: https://phet.colorado.edu/en/simulation/density (or) scan the QR Code
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📝 Summary
- Physical quantities are classified into two types: fundamental quantities (length, mass, time, temperature, electric current, amount of substance, luminous intensity) which cannot be expressed in terms of other quantities, and derived quantities which are obtained by mathematically combining fundamental quantities.
- There are seven fundamental physical quantities in SI Units with their corresponding fundamental units: length (metre), mass (kilogram), time (second), temperature (Kelvin), electric current (Ampere), amount of substance (mole), and luminous intensity (Candela).
- Derived quantities like area, volume, speed, electric charge, and density are obtained by multiplying, dividing, or mathematically combining fundamental quantities, and their units are called derived units.
- Area is a derived quantity that measures space on a flat surface, calculated as length × breadth, with the unit being square metre (m²), and can be calculated for regularly shaped objects using specific formulae.
- Physical quantities require both a value and a unit to express their magnitude, such as walking 2 kilometres where '2' is the value and 'kilometre' is the unit for the physical quantity of distance.